Electronic digital differential analyzer



March 27, 1962 F. G. STEELE 3,027,078

ELECTRONIC DIGITAL DIFFERENTIAL ANALYZER Filed Oct. 28, 1955 12 Sheets-Sheet 1 2/ 3/ Aawfi/Lix rfif/fimla W 5? k 32 g? A 30 30 (C0U/ T/xg l? G 7512) (V) Mada -udv dv u da IN VEN TOR.

Floyd 6. .Sfee/e BY March 27, 1962 F. G. STEELE ELECTRONIC DIGITAL DIFFERENTIAL ANALYZER Filed Oct. 28, 1953 L 54 l 53 5e A.c. some;

12 Sheets-Sheet 2 INVENTOR.

Floyd 5, SfQe/e March 27, 1962 F. G. STEELE 3,027,078

ELECTRONIC DIGITAL DIFFERENTIAL ANALYZER Filed Oct. 28, 1955 12 Sheets-Sheet s l--- El INVENTOR.

Floyd 6. Sfee/e BY ATTOE/VE Y March 27, 1962 F. G. STEELE 3,027,078

ELECTRONIC DIGITAL DIFFERENTIAL ANALYZER Filed 001;. 28. 1953 12 Sheets-Sheet 5 O I I O X J L INVENTOR. 15 Floyd 6. Sfee/e kair Marcl l 27, 1962 F. G. STEELE 3,027,078

ELECTRONIC DIGITAL DIFFERENTIAL ANALYZER Filed Oct. 28, 1953 12 Sheets-Sheet 6 IN VEN TOR.

Floyd 6. Shae/e WIMBEW A TT02NEY March 27, 1962 F. G. STEELE 3,027,073

ELECTRONIC DIGITAL DIFFERENTIAL ANALYZER Filed Oct. 28, I953 12 Sheets-Sheet '7 SAN-Q11),

ATTORNEY March 27, 1962 F. G. STEELE ELECTRONIC DIGITAL DIFFERENTIAL ANALYZER l2 Sheets-Sheet 8 Filed Oct. 28, 1953 INVENTOR: Floyd G. Sfee/e BY March 27, 1962 F. G. STEELE 3,027,078

ELECTRONIC DIGITAL DIFFERENTIAL ANALYZER IN V EN TOR.

F/m d 6. Size/e BY ATTORNEY March 27, 1962 F. G. STEELE 3,027,078

ELECTRONIC DIGITAL DIFFERENTIAL ANALYZER Filed Oct. 28. 1955 12 Sheets-Sheet 11 a E INVENTOR:

Floyd 6'. Shae/e BY March 27, 1962 F. e. STEELE 3,027,078

ELECTRONIC DIGITAL DIFFERENTIAL ANALYZER Filed Oct. 28, 1953 12 Sheets-Sheet 12 LINE f: A 2 C o m l @nn:

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TO OSCILLOSCOPE X TRIGGER cuzcu/r n wh m VERTICAL AMPLIFIER /o0//0o///o/o//oo00 INVENTOR. F/ 0 :1 C1. Sfee/e I='-T I a B BY y 9* .BQ MEAT OIZNEY United States Patent 3,027,078 ELECTRQNIC DHGHTAL DEFFERENTLAL ANALYZER Floyd G. Steele, La Jolla, Calif., assignor to Digital Corn trol Systems, line, La .lolla, Califi, a corporation of California Filed Oct. 23, 1953, Ser. No. 383,780 66 (Ilaims. (ill. 235-152) The present invention relates to an electronic digital differential analyzer, and more particularly to an electronic digital differential analyzer of a greatly simplified design. Difierential analyzers are utilized to obtain approximate solutions of diiierential equations which may arise, in turn, in the mathematical analysis or simulation of problems encountered in physics, chemistry, engineering, etc., if, for example, a differential equation were written to define the theoretical motion of a physical body with respect to time, it might contain a series of successively higher order derivative terms, all taken with respect to time, and representing velocity, acceleration, etc. Such an equation is or" small value until solved, that is, reduced into an algebraic or graphical form of distance only, considered relative to the independent variable time. With this accomplished, an analysis may then be made to determine the actual movement of the body within specific time limits of interests, the results of which may be used to determine the characteristics of the particular system wherein such motion occurs.

All difierential equations may be divided into two general classes, linear and non-linear. A linear ditferential equation is one having only first degree derivative terms, i.e., no squared, cubed, etc., ones, and in which all coefiicients of the derivative terms are functions of the independent variable. All differential equations failing to meet these requirements are termed non-linear ditferential equations. 7

For example, the equations 2 (sin x)y=0 are non-linear differential equations.

In general, all linear differential equations up to the 5th order may be solved usina purely mathematical approach with the solutions appearing as algebraic equations. in the same way, a certain few classes or types of non-linear equations may be solved to obtain pure mathematical solutions in equation form.

In the event a specific difierential equation fails to come under one of the classes having a known mathemat ical solution, other means exist for securing satisfactory approximate solutions thereof, although not in equation form. Thus, any one of a number of well known numerical approximation methods may be employed, each method requiring the use of a set of different equations, the solution being calculated therefrom in a routine step by step manner for consecutive points along the independent variable axis. Each of these numerical approxiice mation methods involves an extremely large number of individual manually performed calculations and, for some of the more complex differential equations'upward of several man years of work may be required to secure an approximation over the regions of interest. Naturally, the solution obtained will be useful only if each of its multitudinous component calculations is performed without error.

In addition to these manual methods, machines, termed broadly difierential analyzers, have been devised to obtain approximate solutions of both linear and non-linear differential equations. Such machines may be classified into three general types, namely, mechanical, analogue and digital. The mechanical analyzers, of which the Bush and the Bush-Caldwell machines are examples, comprise many excessively large and intricate mechanical components requiring in extreme examples, entire buildin-gs for their housing. The mathematical accuracies of the solutions produced by such machines are determined by the mechanical tolerances held in the construction thereof, and, by maintaining extremely rigorous tolerance limits, an output accuracy of approximately one part in ten thousand has been achieved for the best machines, but then only at an excessive cost.

On the other hand, the analogue differential analyzers, although smaller and less expensive than the mechanical analyzers, produce less accurate approximate solutions owing to the rapid accumulation of error in only a relatively few integrator sections. One cause of this error lies in the fact that the analogue analyzers most basic integrator circuit is the well known resistor-capacitor one, it being capable, however, of integrating only with respect to time, the independent variable. Thus, in order to solve non-linear equations, it is necessary to depart from this electronic circuit and introduce motor-driven potentiometers for generating appropriate electrical signals representing the non-linear terms. Thus, such analyzers contain electro-mechanical components in addition to pure electronic circuits with resulting accuracies of one part in five-hundred being average for such machines.

The most recent advance in the field of differential analyzers has been that in the electronic digital field. Such digital analyzers may obtain approximate solutions, using an equivalent time scale, of approximately fifty times the accuracy of mechanical analyzers and at the same time are comparable to the analog analyzer in size and expense. The first of such digital analyzers to appear commercially was the Maddida, the name having been derived from the descriptive phrase magnetic drum digital differential analyzer. The Maddida, although possessing a considerable number of advantages over either of the other two types of analyzers, suffers from an undue electronic complexity, it containing, for example, nearly tubeenvelopes, excluding the power supply, and almost 1,000 crystal diodes. Other digital analyzers appearing commercially since the Maddida have been even more complex owing to their operating in a binary coded decimal system and having special input and output devices associated therewith. Since proper operation of such digital computers is dependent upon the continued and uninterrupted operation of each of its individual components, such an excessively large number of components renders the reliability of these prior digital machines relatively low.

The digital analyzer of the present invention alleviates to a great extent the circuit complexity of the Maddida type of analyzer; it containing, for example, less than 2.0 tube envelopes exclusive of the power supply and less than 400 crystal diodes. As a result, it not only has a much higher degree of reliability, but is also of much smaller Weight, size and, from the economic standpoint, much cheaper to build and service. Furthermore, the

analyzer of the present invention retains the mathematical advantages noted above for the Maddida when contrasted with the mechanical and analogue analyzers.

Essentially, the analyzer of the present invention comprises a motor driven magnetic storage drum having three channels thereon, these being the timing channel, the short or dz information channel, and the long information channel. The long channel is divided into twenty sections, corresponding respectively to the number of integrator sections of the analyzer of the present invention, each section of the long channel being subdivided into eighty spaces each capable of storing binary bit information therein in the form of a magnetized cell of one of two directions of polarization. In operation, the first space of each section of the long channel is normally blank, that is contains a magnetized cell of a direction of polarization corresponding to a blank or a zero binary value, while the next 40 spaces contain the magnetized cells representing the dy and dx code bits in alternate or interplexed order. In the final 39 spaces of each section are found the magnetized cells representing the binary bits of the r and y number registers, interplexed in alternate spaces.

The timing signal, derived from the timing channel, is applied to a single stage binary counter which counts, so to speak, odd and even, or first and second timing signal intervals. This single stage counter, in turn, is utilized to distinguish between the dx and dy code spaces and between the r and y number spaces, since each dy space, for example, will appear during a first timing interval with each dx space appearing during a second interval.

The short or dz channel is 40 spaces long and hence will recirculate twice during the appearance of each of the 80 space sections of the long channel. The signal representing the output value from each of the 20 integrator sections is recorded as a magnetized cell in a separate space on the dz channel, the cells lying in alternate spaces thereof which, in turn, appear during a second timing interval as counted by the single stage counter. All except one of the remaining spaces appearing during first intervals are blank, the one recorded space containing a magnetized cell of a direction of polarization opposite to that of the blank, termed the m mark, and serving to distinguish or separate the various integrator section output values recorded on the dz or short channel.

The m mark is normally precessed, that is, is shifted over two spaces relative to the second interval spaces during the first recirculation of appearance of the short channel during the passage of each section of the long channel and is then recirculated without change during the second appearance of the dz channel. As will be seen, this permits the immediate recording of the cell representing the output value of each integrator section in its own particular dz space in the short channel without special delay devices, since the short channel space immediately following the final space of any given long channel section will always correspond to that section.

Five separate operations are required in solving a differential equation by the analyzer of the present invention, these being clear, mark, fill, shift and compute, which, with the exception of the clear operation, are determined by the settings of three front panel switches, designated mark, register and compute. During the clear operation, both long and short channels are completely erased, that is, a continuous stream of magnetized cells representing zero binary values are magnetically recorded thereon. During the mark operation, the m mark is recorded on the short channel and, additionally, a magnetized cell having the same direction of polarization as the m mark and hereinafter referred to as a fiducial or T* mark is placed on the long channel to appear in the normally blank first space of one long channel section.

During the fill operation, the dy, dx, r and y spaces, hereinafter referred to collectively as registers of the particular section containing the fiducial mark may be separately filled as determined by the operation switch positions, by manually depressing, in a sequence determined by the values to be placed therein, a pair of pushbuttons, termed l and 0, representing one and zero binary values, respectively. After each register of the long channel section containing the fiducial mark has been filled, then the fiducial mark may be shifted to another section by rotating a shift knob on the computers front panel and the registers of that section filled, as previously, by depressions of the l and 0 pushbuttons. As should be noted here, this shift knob is associated with a visual indicator having thereon the numbered integrator sections such that at all times, a visual indication is had of the particular section containing the fiducial mark. After each section required for solving a given equation has been properlyfilled, the compute operation may be initiated and the approximate solution of the equation obtained.

The compute operation is automatically repeated during the passage of each long channel section and comprises a first sub-operation occurring during passage of the interplexed dy and dx registers during which each dz space value corresponding to each dy register space having a magnetized cell representing a one binary value therein is, in effect, counted into a four stage summation dy binary counter, each dz value of one being effective to increase the count by one binary digit value and each dz value of zero being effective to decrease the count by one binary digit value. As will be later demonstrated, the actual operation of the summation dy counter is quite different from conventional counters, the above defined operation being representative of the results only. Also, during passage of the dy and dx registers, each dz value of one corresponding to a magnetized cell in the dx register representing a one binary value is applied to a single stage binary counter which has an initial value of zero. A dz value of zero corresponding to a marked dx code position effects no change in this counter, the final contents of the counter determining later during the same compute operation whether the signal representing the old or new y number of the long channel section being recirculated is subtracted or added, respectively, to the signal representing the r register number.

Then, following a wait sub-operation occurring during passage of any unused portions of the r and y registers, the integrate sub-operation becomes effective with the signal representing the least significant digit of the summation dy counter number being effectively added to the signal representing the least significant y number digit to form a signal representing the new y number least significant digit, which is recorded again on the long channel. Then the signal representing the new or old least significant y number digit is added or subtracted, as determined by the contents of the single stage counter, to the signal representing the least significant r number digit to form the signal representing the least significant digit of the new r number, it being recorded on the long channel just following the magnetized cell representing the new least significant y number digit. Then, the signal representing the next-to-least significant digit of the summation dy count is added to the signal representing the next-to-least significant old y number digit to form the signal representing the next-to-least significant new y number digit with either it or the signal representing the old next-to-least significant y number digit being additionally or subtractively combined with the signal representing the next-to-least significant old r number digit. These alternate operations of the integrate suboperation are continued during passage of the particular long channel section, with the signal representing the carry digit or overflow resulting from the addition or subtraction of the most significant y and r number digits being recorded in the dz channel space corresponding to that section. Upon the next appearance or recirculation of the long channel section, these newly recorded magnetized cells representing the new y and r number digits will form the old y and 1' number digits for the compute operation taking place at that time.

These three sub-operations constituting a compute cycle of operation are sequentially performed during each long channel section excursion with compute operation continuing until either manually or automatically stopped. If manual stoppage is desired, a halt key may be depressed While an automatic stoppage is brought about whenever the y number represented in a long channel section exceeds in magnitude the capacity of its associated y register. This latter stoppage is required since the magnitude of the y number in each section at any time represents the magnitude of the function to be integrated. Upon overflow of the y register, the y number remaining is in obvious error and by having this overflow stop the computation, the normally ensuing incorrect operation is eliminated.

in addition to a description of the present analyzer and its operation, there is also included a simple read-out technique for use therewith whereby the numerical contents of any particular register may be determined at any time. This enables the operator to obtain the magnitude of the answer at a series of points as the solution is progressively advanced by the operation of the analyzer.

It is, therefore, the principal object of the present invention to provide an electronic digital differential analyzer of a greatly simplified design.

Another object of the present invention is to provide an electronic digital differential analyzer including a rotatable magnetic memory including only a permanently recorded timing signal channel and a pair of recirculating information channels thereon.

Still another object of the present invention is to provide a digital differential analyzer including a cyclical memory channel having all of the r and y information of a plurality of integrator sections endlessly recirculated therethrough.

A further object of the present invention is to provide a differential analyzer including a cyclical memory device having first and second streams of information endlessly recirculated therethrough, the first stream of information including the r and y bits of a series of integrator sections with the second stream being the output bits produced by the series of integrations of the information in the first stream.

A still further object of the present invention is to provide a digital differential analyzer including a rotatable magnetic memory having all of the registers of a plurality of integrator sections serially positioned along a first channel thereof, and having the most recent output bit resulting from each integration recorded along a secondchannel thereof.

A further object of the present invention is to provide a digital differential analyzer including a rotatable magnetic memory having all of the input information for a plurality of integrator sections serially recorded and end lessly recirculated along a first channel thereof, and having the output bits from the integrations performed in the integrator sections serially recorded and endlessly recirculated along a second channel thereof, a portion of the input information for each of the plurality of integrator sections being derived from the output bits recorded on the second channel.

A still further object of the present invention is to provide a digital differential analyzer including a rotatable magnetic memory drurn having a pair of recirculating information channels thereon, wherein the input information for 1st, 2nd, 3rd nth integrator sections are serially recorded and recirculated on one of the channels and the output signals of said integrator sections are serially recorded and recirculated on the other of the channels.

Still another object of the present invention is to provide a'digital differential analyzer including a rotatable magnetic memory drum having a first channel divided into 1st, 2nd, 3rd nth sections, one for each inte grator section, each of said sections being 412 spaces in length, with the output signals resulting from the integrations in said 1st, 2nd, 3rd nth integrator sections being recorded and recirculated in alternate spaces on another channel which is 211 spaces in length.

A further object of the present invention is to provide a digital differential analyzer including a magnetic memory device having first and second channels of 411 and Zn spaces, respectively, in length, wherein all of the input information for each of said 1st, 2nd, 3rd nth integrator sections are recorded and recirculated on said first channel in 4n spaces thereof and wherein the output signals resulting from the integrations in said 1st, 2nd, 3rd nth integrator sections are recorded in alternate spaces on said second channel.

Still another object of the present invention is to provide a digital computing device including a cyclical memory storage medium having only two streams of recirculating information for integrating a function u(v) with respect to an independent variable dv.

A further object of the present invention is to provide a device including a cyclical memory storage device for integrating a function u(v) with respect to an independent variable dv wherein binary information representing the value of the function u(v) is stored as the consecutive place digits of a first binary number in the memory device and is continually combined with other information stored in the memory device, in accordance with the consecutive values of dv.

Another object of the present invention is to provide an electronic digital integrating device having a rotatable ma netic storage means wherein a y signal representing the value of the function to be integrated is stored on the stora e means, said y signal being modified at predetermined intervals in accordance with the change of the function which it represents, said y signal being combined at predetermined time intervals with an r signal stored in said storage means to produce a resultant signal representing the result of a mathematical operation upon the numbers representing the result of a mathematical operation upon the numbers represented by the r and y signals, the signal representing the overflow digits from the mathematical operation corresponding to the integral of the function.

A still further object of the present invention is to provide a device for integrating a function u(v) with respect to dv, where du(v, dv) appears as an input electrical signal representing a series of negative and positive counts, and dv appears as a series of signals having first and second values re resenting a series of positive and negative directions of integration, respectively, the device including a cyclical storage means having first and second channels wherein the sum of the counts of the input signal representing du(v, dv) is stored as a first binary signal on the first channel, the device acting in response to the first and second values of each of the dv signals for combining the first binary signal with a second binary signal stored on the first channel for either adding or subtracting, respectively, the first number and the second number represented by said signals and storing a signal representing the most recent carry digit produced by each addition or subtraction on the second channel whereby the series of consecutively stored signals representing the carry digits i also represents u(v)dv.

Another object of the present invention is to provide a digital differential analyzer having all of the input information of a series of integrator sections recirculating around a corresponding series of sections of a first information channel on a magnetic memory drum, only one of said channel sections containing a fiducial mark, the particular section containing the fiducial mark being visually indicated.

Another object of the present invention is to provide a digital differential analyzer having first and second memory channel of Zn and 411 binary digit spaces in length, respectively, wherein, after erasure of the first and second channels, a mark is recorded in one space of the first channel and a second mark in one space on the second channel, and the input information for an integrator section is recorded in the first (4n-l) spaces following the second mark on the second channel.

Still another object of the present invention is to provide a digital differential analyzer including at least one recirculating memory channel having a series of sections and having the input information for a corresponding series of integrator sections recorded thereon, the first space in each except one of the sections of said channel being normally blank with the one section having a signal representing a binary one value recorded in its first space, the signal recorded in the first space of the one section being selectively transferrable to any of the other sections for readout or fill purposes.

A further object of the present invention is to provide an electronic digital computing device having a plurality of operations, each of the operations having a series of automatically sequentially performed sub-operations.

Still another object of the present invention is to provide a digital computing device having a plurality of manually selectable major operations, each of the major operations comprising, in turn, a series of sequentially performed sub-operations, each of the sub-operations being automatically initiated from the previous sub-operation.

A further object of the present invention is to provide an electronic digital computing device having a plurality of manually selectable operations, each of said operations, upon being initiated, comprising a series of sequentially performed sub-operations, each of the sub-operations being ordered by the conduction state sequence of a series of programming flip-flops; the conduction state sequence being changed at the conclusion of each sub-operation to produce the next sub-operation.

A still further object of the present invention is to provide an electronic digital computer device including a pair of recirculating information channels on a rotating magnetic memory, and being capable of performing a plurality of operations, wherein each of the operations includes a plurality of sequentially performed sub-operations, the change of each sub-operation into the next being automatically provided by predetermined data occurrences in the recirculating information channels.

A further object of the present invention is to provide an electronic digital computing device having a series of programming flip-flops wherein the conduction state sequence of the programming flip-flops determines at any time the operation performed by the device.

Another object of the present invention is to provide an electronic digital computing device having at least one major operation which comprises, in turn, a series of sequentially performed sub-operations wherein each of the sub-operations determines the computational manipulations performed on the recirculating information contained in an associated memory device.

Other objects and features of the present invention will be readily apparent to those skilled in the art from the following specification and appended drawings wherein is illustrated a preferred form of the invention, and in which:

FIGURES l and 2 are block schematic representations of a typical integrator section within the differential analyzer according to the present invention;

FIGURE 3 illustrates the interconnections between a group of integrator sections for solving a specified differential equation;

FIGURE 4 illustrates the magnetic memory unit in perspective view and associated computer unit in schematic form of the analyzer according to the present invention;

FIGURE 5 is a circuit diagram of the timing signal source;

FIGURE 6 is a group of signal waveforms illustrating the principles involved in the operation of the circuit of FIGURE 5;

FIGURE 7 is the circuit diagram of a typical writing unit;

FIGURE 8 is a group of signal waveforms appearing in the circuit of FIGURE 7;

FIGURE 9 is the circuit diagram of a typical reading unit;

' FIGURE 10 is a group of signal waveforms appearing in the circuit of FIGURE 9;

FIGURE 11 is a schematic representation of the front panel switching and associated computer flip-flops of the analyzer according to the present invention;

FIGURE 12 is a partly perspective and partly electrical schematic representation of a portion of one of the front panel switches of FIGURE 11;

FIGURE 13 is an electrical schematic representation of one of the front panel pushbuttons of FIGURE 11;

FIGURES 14 through 18 are schematic presentations of the diode gating network associated with the various computer flip-flops;

FIGURE 19 illustrates the information arrangements on the three magnetic memory channels;

FIGURE 20 illustrates the information placement Within a typical integrator section according to the present invention in association with the information on the timing and short channels;

FIGURE 21 is a flow diagram illustrating the flip-flop conduction state sequences and sub-operations performed during each of the major operations of the analyzer according to the present invention;

FIGURE 22 illustrates the informational fiow pattern occurring during portions of a fill dy operation on both of the memory information channels;

FIGURE 23 is a view, partly in perspective, of the indicator mechanism on the front panel;

FIGURE 24 illustrates the informational flow pattern occurring during portions of a short channel precession operation;

FIGURE 25 is a schematic representation of the gating network utilized for deriving output data from the analyzer of the present invention; and

FIGURE 26 is the face of an oscilloscope showing the visual representation of output data produced by the circuitry of FIGURE 25.

(1) COMPUTATIONAL THEORY Referring now to the drawings, there is illustrated in FIGURE 1, a typical integrator section 29 which forms the basic unit of the analyzer of the present invention, it being here shown for the purpose of example only, in a block diagrammatic form. Integrator section 29 includes a counting register 30 for storing a series of electrical signals representing a y number, an accumulating register 31 for storing a series of electrical signals representing an r number, and finally a transfer medium 32 operable in response to an electrical signal representing each input dx value for ordering the additive transfer of a series of electrical signals representing a function of the y number in register 30 to the series of signals in accumulating register 31 but without changing the signals stored in register 30. Registers 30 and 31 contain an identical number of stages, 19 in the present invention, with the y number represented by the signals in register 30 being continuously increased or decreased in magnitude, as by counting, in accordance with an electrical signal representing a dy input. The series of overflow or carry digits produced by the series of dx ordered additive transfers from the most significant digit stage of accumulating register 31 are represented by a dz output signal from register 31.

Considered now in broad mathematical terms, and referring to FIG. 2, if the dy input count is made to represent the differential of a function of a dependent variable u, or equationwise, if dy=du( v, dv) then the count represented by the signals accumulated in the y register at any time is equal to the integral of this functions differential, or u(v).

in the same way, if the information applied to the a'x input represents the differential dv of an independent variable v, then the number represented. by the signals in the register prior to the first carry or overflow digit signal dz represents the integral with respect to dv of the function u (v) stored in the y register or the integral of u(v)dv. However, since the overflow values constituting dz are utilized and not the r number in the register 31, these dz overflow digit signals represent or correspond to the differential of the integral stored in the r register, or merely u(v)dv. As will be noted, the output expression u(v)dv may be either positive or negative in accordance with the sign desired therefor, the method for selectively determining the sign being later presented in more detail.

Turning now to mathematical derivation of the specific integration process performed by typical integrator section 29, it is first necessary to note that the analyzer of the present invention operates on binary numbers less than one and that the most significant place digit represented in counting register 30 designates the sign of the y number, the values of one and zero particularly designating positive and negative y number values, respectively. The binal point is assumed to exist immediately following this most significant or sign digit with the contents of the y or counting register immediately following the ith dy input signal being expressible by:

( -l-yi) (Equation 1) From the above, it will be immediately noted that negative numbers are represented in the y register as the ones complement of the y, absolute magnitude.

The input values represented by the dx signals may be either positive or negative in value in accordance with the above noted independent variable dv values, with transfer medium 32 responding to each signal representing a positive dx value for adding:

w /2 (1+y (Equation 2) to the number represented in r register 31. The term /2 occurs in Equation 2 since the binal point is assumed to exist in the r register immediately to the left of the most significant digit represented therein and hence each of its digits is of one-half the binary value of each of the corresponding y number digits, a further assumption being that the y number is to be added, digit by digit, to the corresponding r number digits. Also, for signals representing negative dx values, transfer medium 32 acts to additively transfer the 2s complement of the number represented in the y register, as exemplified by Equation 1, to the number represented in the r register. This may be represented by:

1= 1)l= yi) (Equation From Equations 2 and 3 it is seen that the general transfer expression for both positive and negative dx value may be written as:

where R represents the contents of the r register follow- (Equation 4) (Equation 5) dzq=2dz-l (Equation 6) From the above, it is readily seen that when dz, equals zero, then du will correspond to a 1 as desired.

Considering now, Equations 5 and 6, the following relationship may be written:

It is seen then that the sum of the dus will approximate the first term of Equation 7 leaving an error of 2(R0Rn).

Now, since the first term n Zyi i i=1 of Equation 7 corresponds to an approximation of the expression:

Indm 1/ where (Equation 8) i=1 it is seen that if the above described operations are performed by an integrator, then such operations produce an approximate integration, recalling that the output dz values are treated as +1 and l magnitudes.

It should here be noted that the operation of the present analyzer does not correspond in exact detail to the mathematical derivation set forth broadly above, its operation actually being, in practice, more accurate. The diiference comes about in that either (l+y,) or (l-l-y, is available for combination with the r number at the end of the ith interval and the machine operates to add (l+y and subtract (1+y to the 1' number in response to positive and negative a'v values, respectively. The reason for this lies in the fact that, recalling that the consecutive y number values represent consecutive values of the function u(v), if the function should double back on itself, as caused by a negative dv value, along the independent variable axis then, upon each doubling back, the final y number value or (1+y prior to the direction switch, should be subtracted after the switch in order that the curve represented by the consecutive y number points he retraced exactly without accumulation of end point errors. The precise mathematical derivation of this operational feature leads to a much greater complexity than the more general treatment given above and for this reason is not here included.

The approximate solution of a specified differential equation may be obtained by properly interconnecting a group of integrator sections of the type illustrated in FIGURES 1 and 2 assuming, of course, that the initial conditions have been properly inserted therein as magnitudes of r and y numbers. By way of example, there is illustrated in FIGURE 3 the interconnections necessary (Equation 9) d(%)=sin udu-udv (Equation 10) Recalling that, from FIGURE 2, a da input accumulates in a y register as u(v), it is apparent if du d were applied as a dy input to an integrator section, the y register contents thereof would represent ii dv However, from Equation 9, it is seen that equals-sin uduudv and hence if sin udu-udv were applied instead as the dy input, the y register contents thereof would correspond t and the solution of the equation, that is, u, could be readily accumulated as the y number in another integrator section. Accordingly, referring to FIGURE 3, an integrator section 34 therein has applied to its dy input, information representing the equation sin udu-udv, which, in turn, is generated, in a manner to be shortly seen, by other integrator sections.

By applying a dv function to the ax input of section 34, the dz output values generated thereby correspond to du which, in turn, is applied to the dy input of another integrator section 35. This dv input function, representing the independent variable, is generated by the integrator section 36 and comprises, for this example, a series of consecutive plus one values ordering the y register contents continually added to the r register contents. The manner in which section 36 produces these consecutive output plus one values will be set forth later in the present disclosure.

The contents accumulated in the y register of section 35 thus represents u, the approximate solution of the equation. Since one of the terms in the equation of the dy input to section 34 is udv, section 35 may be made further use of, in this example, by applying both the :dv function from section 36 and, further, a sign reversing input, here designated as generated on the dz output of an integrator section 37. The manner of generatmg this sign reversing function as well as the meaning thereof will be described later in the present disclosure. Thus, with both dv and applied as the dx inputs to integrator section 35, its output generated function will be udv which, in turn, is applied as one of the input functions to the dy input of the first noted integrator section 34.

The remaining function to be generated is sin udv which is accomplished by a pair of integrator sections 39 and 40. First of all, recalling that d(sin u) =cos udu, if cos udu, neglecting for the moment how such a function is generated, were applied to the dy input of section 39 then sin a would be accumulated in the y register thereof. Now, if the du output function of the section 34 is applied to the dx input of section 39, along with the function generated by section 37, the dz output values thereof would be sin udu which, in turn, com- 12 prises the remaining required input function for the dy input of section 34.

The remaining input function needed for section 39 is cos udu which is obtained by recognizing that d(cos u) and applying the dz output of section 39 or sin udu to the dy input of an integrator section 40 and the du function from section 34 to the dx input thereof. The dz output values of section 40 thus represent cos udu which are applied to the dy input conductor of section 39, thus completing the required interconnections.

From the description of operation of sections 39 and 40, it is seen that both sine and cosine functions may be continuously generated by proper interconnections therebetween. It is, of course, necessary to place a proper initial relationship between the two functions in the respective y registers which may be readily accomplished by, for example, having one y number equal to one with the other y number equal to zero.

(2) MAGNETIC MEMORY DRUM AND CIRCUITRY In the discussion thus far presented in connection with FIGURES l, 2 and 3, each integrator section has been viewed as a separate entity and was accordingly represented by a separate block diagram. In actual practice, however, the analyzer of the present invention has no individual units which constitute an integrator section as a separate and distinct physical entity, as in the case in mechanical and electronic analog differential analyzers.-

Instead, all of the information required in the integrator section of FIG. 1 or FIG. 2 is stored on a cyclical storage device, while all of the arithmetic and control operations and suboperations to be performed on the information are effected by electrical signals from a plurality of gating circuits which, in turn, receive their control signals from a plurality of mechanical and electronic control switches. In particular, according to a preferred embodiment of the analyzer of the present invention, the cyclical storage device is a rotatable magnetic drum having a long information channel and a short information channel. The long channel is divided into a number of sections, corresponding to the number of integrator sections of the analyzer, each section being capable of storing all of the information of the y and r registers of FIG. 1 plus all of the code information necessary to identify the dx and dy information, as set forth in greater detail below. The short channel stores the most recent dz output information from all of the integrator sections of the analyzer and has a bit capacity at least equal to the number of integrator sections.

The information on both channels, considered now in general terms, takes the form of a series of magnetized cells on the surface of the drum representing binary digit values, each value being represented by the direction of orientation or polarization of the magnetic particles constituting the cells. Upon rotation of the drum, considering for the moment either channel, these mag netic particle orientations of each cell, in passing a read point, are transformed by suitable transducers and electronic circuitry into a corresponding electrical signal whose potential magnitudes represent the magnetically recorded values originally producing them at the read point. As defined, the cell representing a recorded binary one value will produce a relatively high potential magnitude in this electrical signal with a recorded zero value producing a relatively low potential magnitude.

These serially appearing electrical signals from the read point are then acted on by the computer in accordance with the particular type of operation being performed at that time with each signal magnitude either being changed into the other magnitude or remaining the same as determined by the mathematical result or answer of the operation. Then, these serially appearing answer signals are applied by an appropriate transducer 13 and circuitry to a write point adjacent the same channel to be magnetically recorded thereon, again as the magnetized cells previously defined. This newly recorded information is then, owing to the rotation of the drum, delayed in time until it reappears at the read point with subsequent transformation into signal form, computational operation thereon, and re-recording on the channel.

The amount of delay afforded each binary bit of the information on a channel is determined by the length of its associated channel and angular velocity of the drum, each channel length being determined, in turn, by the arcuate spacing between the read and write points measured around the drums periphery. Also, the total information found on each channel will make one complete excursion each instance the drums surface moves a distance corresponding to the spacing between its corresponding read and write points. In addition, since the drum will, in practice, be continuously rotated, the information on the two channels will be endlessly recirculated and each bit thereof be made available time and time again for individual computational operations.

In addition to the above noted long and short memory channels in the present analyzer, there is permanently recorded on another or third channel extending completely around the drum, timing information which is represented by a repetitive alternate alignment pattern of the drums magnetic particles. By sensing this timing information at a read point with an appropriate transducer and electronic circuitry, a corresponding timing or clocking signal is produced, it being of an endlessly repetitive nature, similar to its corresponding magnetic pattern, and serving two basic and fundamental functions in the operation of the analyzer.

First of all, the timing signal is employed as a measuring function or time scale for the information recording operation on the two recirculating channels by timing the recording of each bit thereon and hence alloting thereto a discrete channel space. This serves to eliminate any possible scrambling or intermingling between consecutive pairs of such bits with the result that all binary bits thus recorded will be distinctly recognizable upon appearing at the read points.

Also, the clocking signal provides for synchronous and time related operations between all of the electronic switches included in the present analyzer and additionally produces, as will become obvious later, the serial transfer, if called for, of the digit contents represented by the conduction state of one electronic switch into another similar switch.

Turning now to the specific magnetic memory arrangement employed in the present analyzer, there is illustrated in FIGURE 4, a magnetic memory unit 42 shown in conjunction with a computer unit 43, here illustrated in block schematic form, units 42 and 43 forming the two basic components of the present analyzer. Memory unit 42 includes a source of driving energy, such as an electric motor 45, preferably of an alternating-current synchronous type, driving a cylindrically shaped memory Wheel or drum 46 in the designated counter clockwise direction of rotation at an angular velocity of, for example, 1800 revolutions per minute. Drum 46 is preferably formed of plastic or other non-magnetic material and includes a thin coating 48 of magnetic iron oxide extending around its outer edge or periphery. A stationary circular mounting ring 49, having a greater diameter than that of drum 46, is positioned between motor 45 and the drum and serves as a mounting support for various transducer elements to be shortly described.

A plurality of magnetic transducer mounting brackets, are secured to mounting ring 49 and contain appropriately sized openings for mounting associated magnetic heads, the combined brackets and heads functioning as read, write and erase units. Particularl a read unit 52 and a write unit 53 are mounted in circumferential alignment around drum 46 to sense and record data, respectively, on the long information channel 54 while a read unit 56 and a write unit 57 are positioned relative to drum 46 so as to sense and record data, respectively, on the short information channel 58. An erase unit 5% is also illustrated, it including a pair of permanent magnets aligned adjacent tracks 54- and 58 so as to erase the information previously recorded thereon by write units and 57, the erasure taking the form of orientating the magnetic particles of tracks 54 and 53 in a direction parallel to the direction of their movement to correspond to a continuous series of binary Zero digit values.

Finally, a read unit 61 is illustrated, it serving to sense the permanently recorded timing data on a timing track 62, likewise circumferentially recorded around drum 46. Additional details of the read, write and erase units will be found later in connection with the description of certain electronic circuitry associated therewita. As is also indicated, read units 552 and 56, and at deliver their output signals to computer unit 43 while write units 53 and 57 receive input signals from unit 43.

Computer unit 43 includes appropriate electronic circuitry for both converting the magnetic information sensed by the read units from the long, short and timing signal channels into the before mentioned electrical signals and for supplying electrical signals representing output computational information to the write units for their subsequent recordment on the respective short and long channels to complete the recirculation process.

Considering first the electronic circuitry tassociated with the timing read unit at, reference is made to FIG- URE 5 wherein unit 61 is shown to include a magnetizable core 66, preferably of ferrite material, of a horseshoe or C-shaped configuration and having a coil 67 wound thereon. The narrow air gap between the core pole faces is positioned perpendicularly to the travel of channel 62 with one end of coil 67 being capacitively coupled to the grid of a first triode 76, within an amplifier unit 69, the other end of the winding being connected to ground. A grid resistor is coupled between the grid of triode Iii and ground, it being of a sufficiently high value to furnish a suitable contact potential for linear operation of the triode. The cathode of triode 70 is grounded while its anode is coupled through a conventional plate resistor to the terminal E of a source of, for example, 250 volts positive potential, the source not here being illustrated.

The anode of triode 70 is capacitively coupled to the grid of a second triode 75, also within amplifier 69, the grid of triode 75 being connected through a conventional grid resistor to the junction point between a pair of resistors 72 and 73, in turn, serially connected between a terminal E of a source of, for example, 350 volts negative potential, and ground. The anode of this second triode is connected through a conventional plate resistor to terminal E with the output signal of the amplifier unit being capacitively coupled from the anode of triode 75 to the input terminal of an unistable or one-shot multivibrator circuit '73.

In particular, the input terminal of circuit 78 is coupled through a resistor 79 to a terminal E of a. source of, for example, 5 volts positive potential, and is additionally coupled through the anode to cathode of a diode 80 to the grid of a first triode 82. The cathode of triode 82 is grounded, the anode thereof being coupled through a plate resistor to terminal E and through a pair of diodes 83 and 84 to the terminals E and E respectively, of a pair of clamping potential sources producing substantially constant positive output voltages of, for example, 14-0 and 10-0 volts, respectively. The magnitudes of these terminal E and E potentials are extensively referred to, hereinafterwards, as low and high voltage levels, respectively. 1

This triode 82 anode is also coupled through a coupling capacitor 86 to the grid of a second triode 88, the grid thereof being also connected through a variable resistor 92 to terminal E The cathode of triode 88 is grounded and its anode is both coupled through a plate resistor to terminal E and through a paralleled resistor 89 and capacitor 90 to the grid of triode 82 and from there through a resistor 91 to terminal l inally, its anode is connected through diodes 93 and 9. to the pair of clamping potential terminals E and E respectively.

The output signals of circuit 78 appetr on the anodes of triodes 82 and 88, respectively, the output signal of triode 82 being applied to the C input conductor of an electronic switching device, or bistable multivibrator circuit, such as flip-flop C1. This S conductor is, within flip-flop C1, coupled through a capacitor 97 and diode 98 to the grid of a first triode 100. The other output signal of circuit 78 is applied to the Z input conductor of flip-flop C1 which, in turn, is coupled serially through a capacitor and diode, corresponding to capacitor 97 and diode 98, respectively, to the grid of a second triode 102. The common junction between capacitor 97 and diode 98 is coupled through a resistor to terminal E as is the common junction between the capacitor and diode within the grid circuit of triode 102.

The grids of triodes 100 and 102 are coupled through a pair of grid resistors to terminal E while their anodes are coupled through a pair of plate resistors to the E terminal. Both anodes are likewise clamped as previously described for circuit 78 by appropriate diodes connected to the E and E terminals. in addition, the anode of triode 100 is coupled through a paralleled resistor-capacitor combination 104 to the grid of triode 102 while the anode of triode 102 is coupled through a paralleled resister-capacitor combination 105 to the grid of triode 100. Finally, the output signal cl of flip-flop C1 is derived from the anode of triode 100 while its so-called complementary output signal, designated cl is taken from the anode of triode 102.

For explaining the operation of this timing read circuit, reference is made to FIGURE 6 wherein is set forth a group of signal waveforms appearing at various points in the circuit. First illustrated, in a schematic manner, is the permanent magnetization pattern on timing signal channel 62, the pattern comprising an endless series of substantially equally lengthened magnetized areas or cells, the magnetic oxide particles in consecutive areas being orientated in alternate directions parallel to the direction of track travel. Upon passage of channel 62 beneath the pole faces of core 66, alternate positive and negative pulses are produced across coil 67, the two polarities corresponding to the two changes of direction of the flux pattern. These pulses of alternate positive and negative polarity are illustrated by the signal waveform, generally designated 107, and after amplification by triode 70, appear as negative and positive pulses, respectively, on its anode.

Resistors 72 and 73, coupled between ground and terminal E serve as a voltage dividing network to apply a negative bias of such magnitude to the triode 75 grid to operate the tube within its linear operating region. Thus, the positive and negative'pulses appearing on the anode of triode 70 are amplified by triode 75 to appear as corresponding output negative and positive pulses, respectively, as exemplified in the signal waveform designated 108 in FIGURE 6.

Considering now for the moment, multivibrator 78, the grid current drawn through variable resistor 92 from triode 88 establishes a zero valued grid potential with a normal full conduction status resulting therefrom. With triode 88 fully conducting, its anode potential is determined by the 100 volt clamping potential appearing on terminal E and, with this magnitude of potential appearing on the anode, the relative values of resistors 89 and 91 are such as to apply a sufiiciently negative bias to the grid 16 of triode 82 that it assumes a cut off condition with its anode potential being that on terminal E or 140 volts.

Now, each positive pulse appearing on the anode of triode 75 is capacitively coupled through diode to the grid of triode 82 resulting in a rise of its grid potential and flow of plate current. This, in turn, causes a reduction of its anode voltage which, when coupled through capacitor 86 to the grid of triode 88 results in a reduction of its normal full anode current flow and increase of its anode potential. This increase in potential is, in turn, coupled by way of resistor 89 and capacitor 90 to the grid of triode 82 which, in turn, raises its grid potential still further with a subsequent decrease of its anode voltage.

This interaction between the two triode circuits continues in a substantially instantaneous manner until their normal conduction states have been reversed with triode 82 fully conducting and triode 88 non-conducting, their respective anode potentials beingregulated by the E and E terminal clamping voltages, respectively. These conduction states are maintained until the negative charge accumulated on capacitor 86 during the triggering operation gradually leaks off through resistor 92 and the triode 88 grid potential is raised by the positive potential on terminal E above cut off and triode 88 begins to again draw current. This, in turn, causes a reversal of the previously described action in that the current drawn through triode 88 causes a decrease in its anode potential, which when coupled to the grid of the then fully conducting triode 82 causes a reduction in its anode current and rise in its anode voltage. This interaction continues, again in a substantially instantaneous manner, until the conduction states of the two triodes are once more back to their first described or normal condition.

The time that the reversed conduction states takes place is determined primarily by the relative values of capacitor 86 and variable resistor 92 which, in turn, are preferably adjusted such that the reversed conduction state takes place for substantially the interval of time that each magnetically aligned area of track 62, producing an initial positive pulse, is passing beneath head 61. Thus, in FIGURE 6, the waveform generally designated 109 is that appearing on the triode 82 anode with the signal appearing on the triode 88 anode being designated by the waveform designated 110. It is seen, in accordance with the above explanation, that waveforms 109 and 110 are complementary with respect to each other, that is, when the first is at the clamped high potential level, the second will be at the low clamping level, and, alternately, when the first is low, the second will be high.

Considering now the operation of flip-fiop C1, signal 109 is applied across capacitor 97 and resistor 99, the two acting as a differentiating circuit to produce positive and negative going pulses for each change of signal 109 in the positive and negative directions, respectively. Since diode 98, owing to the direction of its connection, will effectively block all positive pulses, only the negative pulses will be conducted therethrough to appear as a voltage drop across the triode 100 grid resistor. These negative pulses, as they appear at the grid of triode 100, are illustrated by the signal Waveform, generally designated 111 in FIGURE 6.

In the same way, the grid input circuit to triode 102 serves to differentiate signal with only the resultant negative pulses thereof appearing on the grid of triode 102, the grid input signal being represented in FIGURE 6 as the signal waveform designated at 112.

If triode 100 of flip-flop C1 is initially assumed fully conducting with its output cl signal being accordingly at the E or low voltage level, then the next negative pulse appearing in signal 111 will lower its grid potential causing a decrease in its plate current flow and an increase in its anode potential. This rise in potential is coupled by way of resistor and capacitor combination 104 to the grid of the then cut-oh triode 102. This rise in triode 

